guidance, flight mechanics and trajectory optimization

2.4.3 Optimum Continuous Thrust -- Guidance for: the Final Maneuvers
2.4.3.1 Free Space Node1
In this section, the optimum steering program for the final thrust period
of a rendezvous is developed. That is, it'is assumed that the two vehicles
have been established on nearly identical trajectories such that the range
rate is negative and the range is large enough that rendezvous can be
accomplished without overshooting the target. A single thrust period will
accomplish the maneuver, but depending on when the maneuver is initiated, a
coast period may be required before thrusting begins. A switching function,
however, is not developed in the cour.se of the optimization as was done in
the last section. Rather, the time to initiate the thrust maneuver is
determined after the optimum steering program for the thrust period has been
determined. Although the simplifications necessary for this derivation are
much more restrictive than those of the last section, the result is a closed
form expression for the steering function in terms of the initial conditions
of the relative motion. In contrast, the thrust components of the previous
section were given in terms of a two-point boundary value problem whose
solution is analytically intractable. Because of the closed form of the
solution, it is applicable for use in an on-board guidance system whereas the
previous method is probably not because of the two-point boundary value
problem whose solution must be obtained by iterative techniques which are
often slow to converge.
The coordinate system in which the problem is considered is two-dimen-
sional, non-rotating, and fixed to the target vehicle. The X axis is in the
direction of the relative velocity vector between the two vehicles at the
start of the rendezvous maneuver.
The rendezvous vehicle is assumed
to have a constant
thrust motor
which provides a
constant acceleration
a,. The equations of motion are:
where 8 is the angle between the thrust vector and the X axis. The problem
is to find 8 as a function of time so that the fuel required for the rendezvous
maneuver is minimum. Since the constant thrust motor is operating con-
tinuously, the amount of fuel required is proportional to the time that the
motor is operating. Therefore, minimizing the fuel is equivalent to mini-
mizing the time of burning, i.e., the problem is
find 8 (t) OLf Lt,
such that tf is minimum and x (t,) = i (t,) = 0. The optimizations will be
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